Determination Methodology for Stability Domain of Hybrid Electric Power System with Multiple Excitation Sources

Irregular internal excitation (engine excitation and motor excitation) and external load excitation can cause torsional vibration in hybrid electric propulsion system which even leads to the break of shaft. The objective of this paper is to determinate the stability domain of hybrid electric power system under different drive mode with multiple excitation sources. To achieve the goal, the simplified three-mass torsional model of hybrid electric power system is established. Then we apply the multi-scale method to solve the nonlinear torsional model. Finally we set different parameters of engine speed to simulate torsional vibration characteristics of hybrid electric power system. The simulation results exhibit that the torsional vibration characteristics of hybrid electric power system under two drive mode are different because the directional relationship of the moment of engine and permanent magnet synchronous motor (PMSM) is distinctive. It’s hard to eliminate the torsional vibration of hybrid electric propulsion system due to phase difference between the engine excitation and the load excitation.


Introduction
In recent years, deteriorating energy and environmental problems promote the development of electric power systems and hybrid electric power systems [1,2]. Nevertheless, the application of electric power systems have encountered shortcomings, such as long charging time, limited operating range and lifespan. On the other hand, hybrid electric power systems have been applied increasingly for their flexible working mode, wide operating range and low emission [3,4,5]. Usually, the hybrid electric power system is composed of two power sources: one is the combination of engine and generator, and the other is the combination of motor and battery [6,7]. However, due to the complexity of the configuration of hybrid electric power system and strong nonlinearity of engine and permanent magnet synchronous motor(PMSM), irregular engine excitation and electromagnetic excitation can cause torsional vibration which even leads to the break of shaft [8,9]. So it's essential to analyze the torsional characteristics of hybrid electric power system to determine the stability domain with the engine and PMSM participated in.
The hybrid electric power system is a multi-degree of freedom(DOF) nonlinear system with multiple excitation sources. It is considered that the mass distribution of the hybrid electric power system is lumped [10]. Tang et al. [11] established a 16-DOF torsional vibration model of hybrid electric vehicle power system. The model is applied to analyze the torsional vibration characteristics and give the theoretical prediction of the natural frequency and corresponding vibration modes. The dynamic model was also used to present the torsional modes of hybrid electric power system in hybrid driving mode. The results indicate that the low frequency vibration is concentrated on the vehicle and wheels. Engine noise is the main noise in the hybrid driving mode. Tang et al.
[12] developed a simplified three-mass torsional model of the hybrid electric power system to determine the dominant frequency. The results show that the simplified model can be used to describe the low frequency vibration characteristics of the hybrid electric power system accurately. However, the simplified three-mass model concentrates on the unbalanced torque of engine and does not consider the electromechanical coupling relationship between the engine and motor. Chen et al. [13] established the two-mass torsional vibration model of hybrid electric power system considering electromechanical coupling. Yue et al. [14] specifically analyzed the amplitude-frequency characteristic of torsional vibration in two different modes of the hybrid electric power system (boosting mode and generating mode). The corresponding results of the engine were compared to reveal that the coexistence of electromagnetic torque of motor and the engine torque changes the torsional vibration characteristics of the original engine shafting and intensifies the amplitude of the torsional vibration. Zuo et al. [15] built a multi-body dynamic model based on a simplified structure of a parallel hybrid system. The incentive model of engine was established in MATLAB, the dynamic simulation results show that the change of engine cylinder pressure is the main reason that affects the torsional vibration of the shaft system. However, many researches focus on the influence of engine excitation, as well as the analysis of natural frequencies and corresponding modalities of hybrid electric power system.
The purpose of this paper is to determine the stability domain of hybrid electric power system under two typical drive mode. A simplified three-mass nonlinear torsional model of hybrid electric power system under various drive mode was developed using the lumped parameter method in Section 2. The first-order approximate solution of the model was solved by multi-scale method in Section 3. The nonlinear dynamics theory was applied to reveal the torsional vibration characteristics of hybrid electric power system. Section 4 introduces the results of the influences of variable engine speed on the nonlinear torsional vibration characteristics of hybrid electric power system. Finally, conclusions are drawn in Section 5.

Electromagnetic torque model of PMSM
Considering the torsional vibration angle of PMSM, the rotor synthesis fundamental wave magnetomotive force is given below [16,17]: (1) Electromagnetic torque of PMSM is given below when the torsional vibration angel is .

(sin sin( ) cos cos( ))
; R is inner radius of stator; l represents effective length of rotor; 0  is air gap permeability of PMSM. The simplified equation of electromagnetic torque of PMSM is shown using Taylor approximation.

Mechanical torque model of engine
The unbalanced force sources of a four-cylinder engine are mainly composed of reciprocating moment of inertia j T and combustion gas moment g T . The combustion gas moment g T can be expressed by  [18,19].
Since the fourth-order and higher-order unbalanced moments are only 2.5% of that of second-order, it is mainly necessary to consider the second-order imbalance torque. ( Where, j m represents reciprocating equivalent mass of piston connecting rod group; r is crank radius; D is piston diameter;  represents connecting rod ratio.
In actual analysis of engine torsional vibration, only the interference torque g T generated by changeable gas pressure is generally considered. Furthermore, the cosine component in combustion gas moment g T is small compared with the sine component. Considering torsional vibration angel, simplified mechanical torque model of engine is given below.

Nonlinear torsional model under boosting mode
A schematic representation of hybrid electric power system is shown in figure 1. The hybrid electric power system gets into the compound driving mode when the clutch is locked, and the PMSM drives, together with the engine [20,21]. The mechanical rotation equations is given below: Where 1 J , 2 J and 3 J represent the moment of inertia of engine, PMSM and load respectively;  represent rotation angles at the end of the two shafts; 1 C and 2 C are damping coefficient; , , The equivalent torsional vibration equations of the hybrid electric propulsion system under the engine torque disturbance is given as below: We substitute the following equations into Eqs.9   22  3 1 2  3 1  2  1  1  2  2  1 1  2  2  2  3  1  1  2  2  2  1  2  2  3  2  2  3 33 , , , ,

Theoretical analysis of stability domain
First the small parameter  and the following scale transformation [22] are introduced.
The two-degree-of-freedom nonlinear model with the combination of parameter excitation and external excitation is shown as follows.   5 We use the method of multiple scales to find the first-order uniform asymptotic solution of Eqs. 12. 1 10 0 1 11 0 1 We introduce two tuning parameters 1  and 2  .
Where, The characteristic equation at the zero solution has the following form: According to the Routh-Hurwitz principle [23], the stability conditions is given as follows.
, and the load excitation frequency is set to be 2 =20  .

Conclusion
In this paper, a three-mass torsional vibration model of engine-PMSM-load of hybrid electric propulsion system is established by using lumped parameter method. Using nonlinear dynamics theory, the influence of different engine speed and load excitation frequency on nonlinear torsional vibration stability of hybrid electric propulsion system is studied. We get the following conclusions: (1) There is a phase difference between the engine excitation and the load excitation in the hybrid electric propulsion system, which makes the hybrid electric propulsion system hard to eliminate the torsional vibration; (2) With the engine speed as the control parameter, the torsional vibration characteristics of the two shaft are different because the directional relationship of the moment of engine and PMSM is distinctive from that of PMSM and load. When